Past Functional Analysis Seminar

19 November 2013
17:00
to
18:30
Tomasz Kania
Abstract
We address the following two questions regarding the maximal left ideals of the Banach algebras B(E) of bounded operators acting on an infinite-dimensional Banach space E: i) Does B(E) always contain a maximal left ideal which is not finitely generated? ii) Is every finitely-generated maximal left ideal of B(E) necessarily of the form {T\in B(E): Tx = 0}? for some non-zero vector x in E? Since the two-sided ideal F(E) of finite-rank operators is not contained in any of the maximal left ideals mentioned above, a positive answer to the second question would imply a positive answer to the first. Our main results are: Question i) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; Question ii) has a positive answer if and only if no finitely-generated, maximal left ideal of B(E) contains F(E); the answer to Question ii) is positive for many, but not all, Banach spaces. We also make some remarks on a more general conjecture that a unital Banach algebra is finite-dimensional whenever all its maximal left ideals are finitely generated; this is true for C*-algebras. This is based on a recent paper with H.G. Dales, T. Kochanek, P. Koszmider and N.J. Laustsen (Studia Mathematica, 2013) and work in progress with N.J. Laustsen.
  • Functional Analysis Seminar
12 November 2013
17:00
to
18:12
Martin Kolb
Abstract
We study the inuence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the flat case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the flat case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation.
  • Functional Analysis Seminar
22 October 2013
17:00
to
18:25
David McConnell
Abstract
The problem of representing a (non-commutative) C*-algebra $A$ as the algebra of sections of a bundle of C*-algebras over a suitable base space may be viewed as that of finding a non-commutative Gelfand-Naimark theorem. The space $\mathrm{Prim}(A)$ of primitive ideals of $A$, with its hull-kernel topology, arises as a natural candidate for the base space. One major obstacle is that $\mathrm{Prim}(A)$ is rarely sufficiently well-behaved as a topological space for this purpose. A theorem of Dauns and Hofmann shows that any C*-algebra $A$ may be represented as the section algebra of a C*-bundle over the complete regularisation of $\mathrm{Prim}(A)$, which is identified in a natural way with a space of ideals known as the Glimm ideals of $A$, denoted $\mathrm{Glimm}(A)$. In the case of the minimal tensor product $A \otimes B$ of two C*-algebras $A$ and $B$, we show how $\mathrm{Glimm}(A \otimes B)$ may be constructed in terms of $\mathrm{Glimm}(A)$ and $\mathrm{Glimm} (B)$. As a consequence, we describe the associated C*-bundle representation of $A \otimes B$ over this space, and discuss properties of this bundle where exactness of $A$ plays a decisive role.
  • Functional Analysis Seminar
11 June 2013
17:00
to
18:15
Alex Belton
Abstract
Although generators of strongly continuous semigroups of contractions on Banach spaces are characterised by the Hille-Yosida theorem, in practice it can be difficult to verify that this theorem's hypotheses are satisfied. In this talk, it will be shown how to construct certain quantum Markov semigroups (strongly continuous semigroups of contractions on C* algebras) by realising them as expectation semigroups of non-commutative Markov processes; the extra structure possessed by such processes is sufficient to avoid the need to use Hille and Yosida's result.
  • Functional Analysis Seminar
14 May 2013
17:00
to
18:07
Tom ter Elst
Abstract
We consider a bounded connected open set $\Omega \subset {\rm R}^d$ whose boundary $\Gamma$ has a finite $(d-1)$-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator $D_0$ on $L_2(\Gamma)$ by form methods. The operator $-D_0$ is self-adjoint and generates a contractive $C_0$-semigroup $S = (S_t)_{t > 0}$ on $L_2(\Gamma)$. We show that the asymptotic behaviour of $S_t$ as $t \to \infty$ is related to properties of the trace of functions in $H^1(\Omega)$ which $\Omega$ may or may not have. We also show that they are related to the essential spectrum of the Dirichlet-to-Neumann operator. The talk is based on a joint work with W. Arendt (Ulm).
  • Functional Analysis Seminar
9 May 2013
17:00
to
18:10
James Kennedy
Abstract
Almost 50 years ago, Kac posed the now-famous question `Can one hear the shape of a drum?', that is, if two planar domains are isospectral with respect to the Dirichlet (or Neumann) Laplacian, must they necessarily be congruent? This question was answered in the negative about 20 years ago with the construction of pairs of polygonal domains with special group-theoretically motivated symmetries, which are simultaneously Dirichlet and Neumann isospectral. We wish to revisit these examples from an analytical perspective, recasting the arguments in terms of elliptic forms and intertwining operators. This allows us to prove in particular that the isospectrality property holds for a far more general class of elliptic operators than the Laplacian, as it depends purely on what the intertwining operator does to the form domains. We can also show that the same type of intertwining operator cannot intertwine the Robin Laplacian on such domains. This is joint work with Wolfgang Arendt (Ulm) and Tom ter Elst (Auckland).
  • Functional Analysis Seminar

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